[L 1 ,L 2 ,…L n ] Shape regression S(t): R n -&gt

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Shape spaces and metrics in an application perspective

Mads Nielsen

eScience Centre, DIKU, University of Copenhagen

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Overview

The application perspective:

Data and task

Which structure is needed on the models to subserve the

task?

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Disclaimer

Slides have been stolen from Mumford,

Dam, Chennai, and many more

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Osteoarthritis (OA)

OA is a degenerative joint disease in knees, hips, … Effect:

Pain, Reduced range of motion

Rule of thumb:

Age in years gives % chance of OA

Treatment:

Symptom control

Current golden standard:

-Kellgren & Lawrence Index -Joint space width

www.chclibrary.org www.chclibrary.org

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Quantification Framework

Folkesson, Dam et al. 2007 Trans Medical Imaging Dam et al. 2008

Medical Image Analysis Dam, Folkesson et al. 2007

Osteoarthritis & Cartilage Qazi, Dam, Karsdal et al. 2007

Osteoarthritis & Cartilage

Volume Thickness Curvature

Smoothness,

Homogeneity

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Risk of vertebral fractures

Current standard of fracture grading:

Bone Mineral Density based on dual x-ray Our approach:

Statistical shape analysis

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Visualization

Green: Likely to stay intact

Blue: Mean spine shape

Red: Likely to fracture

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Tasks

Classification S —> [L ₁ ,L ₂ ,…L _n ] Shape regression S(t): R ⁿ -> S

Marker regression t(S): S -> R

Prior for segmentation p(S): dist. on S In all cases, a metric on the space S of

shapes S is essential

Finding usefull metrics is non-trivial

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Shapes

Shape = Geometry \ Position

Shape is a qouotient manifold (mayby embedded in Geometry space)

Metric on the geometry space, may be inherited (projected) to the shape space

Kendall : Points in R

²ⁿ

\ Similarity

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Start with a fixed curve parametrized by ( ) Define a local chart near :

( ) ( ) ( ). ( ), ( ) unit normal to ,

image of

a

a a

C

s s

s s a s n s

n s C

C

a r r

f f

y f

y

Î

= +

=

S

The set Σ of all smooth plane curves

forms a manifold!

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Think of Σ geometrically

• A curve on Σ is a warping of one shape to another.

• On Σ ^, the set of ellipses forms a surface:

• The geometric heat equation:

is a vector field on Σ ^∂ ^C _t

^t

= κ

^C^t

ⁿ

^C^t

^,

∂ r

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Advantages of L ² metrics

Have simple notion of a gradient to form flows

Have a beautiful theory of locally unique geodesics, thus a warping of one shape to another.

Can define the Riemannian curvature tensor. If non- positive, have a good theory of means.

Can expect a theory of diffusion, of Brownian

motion, hence Gaussian-type measures and their

mixtures.

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A geodesic in the simple L ² metric

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

θ axis

t axis

The line on the bottom is moved to the line on the top by growing

“teeth” upwards and then shrinking them again.

But distances collapse in this metric:

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Fixes of this bug:

Michor + Mumford: L ₂ + curvature Yezzi: Sobolev metric

Charpiat: Bounded curvature

Sommer: Finite bandwith by resampling Trouve, Younes: Diffeomorphic

Others use Hausdorff metric

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Michor+Mumford

For small shapes, curvature is negative and the path nearly goes back to the circle (= the ‘origin’).

Angle sum = 102 degrees.

For large shapes, curvature is positive, 2 protrusions grow while 2 shrink. Angle sum = 207

degrees.

Hence construction depends on scale

2 2 2

(1 )

a = ∫ + A κ a ds

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Charpiat, Faugeras, Keriven

The shape space S is limited to shapes S where the curvature (the extrinsic curvature of the curve in R ² ) is limited to k < k ₀

This is scale dependent and the ordinary L ₂

geodesic depends on k ₀

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Yezzi:

Geometric Sobolev-type norms;

We define

|h|

²_Sobolev

:= |h|

²

+ λL

²

|Ds h|

₂²

where h : S

₁

-> R

²

is a perturbation of the curve c, L is the length of c,

Ds is the arclength derivative,

This is a negatively curved space.

This construction is independent of scale

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Sommer et al

Later today

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Trouve, Younes

Define a sobolev type metric on flows on the embedding plane.

This introduces also a flow on embedded curves

This was introduced here by Tom Fletcher

It is negatively curved

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Same metric: a reflection of its negative curvature for small shapes: to get from any shape to any other which is far away, go via ‘cigars’ (in neg. curved

space, to get from one city to another, everyone takes the same highway)

Negatively curved spaces

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